I was unsure if I should post this here or on the Mathematics page, so I decided to do both. Here's the problem I want to describe: Suppose I have a sphere, discretely represented by a large set of simple planar polygons in $\mathbb{R}^3$. The polygons cover the surface of the sphere and are adjacent to each other with no (or very slight) overlaps between neighbouring polygons. However, the normals of the polygons are the analytical normal of the sphere at the centroid of each polygon, with a perturbation ($<20^{\circ}$) added.
In my actual application (surface tension modelling), the surface can be an arbitrary, smooth, and closed surface. Therefore, unlike for a sphere, I do not know the local surface normal $\textit{a priori}$. I only have the piecewise-linear, polygonal representation of the surface. In order to reduce the perturbations and achieve a smoother distribution of the polygon normals, I want to represent the problem as a set of massless plates allowed to freely pivot about their area centroids, with tensile springs connecting each plate with each of its immediate neighbours. The springs are attached at the edges of each plate.
Can anyone help me describe this problem mathematically, in terms of an objective function that I can numerically minimise? It's important to note that I don't actually intend to simulate the physics of the plate-spring system. I just want to find its lowest energy state, just so I can compute the polygon normals.
$\textit{p.s.}$ I have tried other discrete methods. But there are still large errors in comparison to the analytical normals. My thinking is that the solution to the plate-spring system will yield the most accurate normals I can hope to achieve.
